Nondegenerate perturbation theory

Perturbation theory for an isolated eigenvalue with a one-dimensional eigenspace. First-order nondegenerate perturbation theory gives the diagonal perturbation expectation, while second-order nondegenerate perturbation theory accounts for coupling to other unperturbed eigenstates through energy denominators. Symmetry can permit its use within one block even if another, uncoupled symmetry block has the same unperturbed eigenvalue.

For the normalized sine eigenfunctions of a particle in a box on $(0,a)$, multiplying the lowest one by $ϵ\cos (2\pi x/a)$ gives $ϵ({\varphi }_3-{\varphi }_1)/2$. Thus the first-order diagonal shift is $-ϵ/2$, and the only second-order coupling is to level three. With $E_3=9E_1$, its contribution is $({ϵ}^2/4)/(E_1-E_3)=-{ϵ}^2/(32E_1)$, giving the displayed expansion. This simple selection rule makes the full second-order perturbation theory sum a single term.

The ratio $|⟨k|V|i⟩/(E_i-E_k)|$ measures a leading admixture between unperturbed eigenstates. Small matrix elements compared with relevant gaps justify a nondegenerate expansion locally. A small overall perturbation does not suffice near a coupled degeneracy, where the nearly degenerate block needs diagonalization.